How to use this calculator
- Type the numerator (top number) and the denominator (bottom number).
- For a mixed number such as 2 4/6, put the whole number in Whole. Leave it empty for an ordinary fraction.
- The answer updates as you type: the fraction in lowest terms, the mixed number, decimal and percentage, the GCF, the full working, equivalent fractions and a picture of the fraction.
Negative numbers and decimals both work, and nothing you type is saved.
How to simplify a fraction (GCF method)
A fraction is simplified, or in lowest terms, when no whole number bigger than 1 divides both the top and the bottom. To get there in one step:
- Find the greatest common factor (GCF) of the numerator and denominator: the largest number that divides both exactly.
- Divide the numerator and the denominator by the GCF.
Worked example: simplify 12/16. The factors of 12 are 1, 2, 3, 4, 6 and 12; the factors of 16 are 1, 2, 4, 8 and 16. The largest one they share is 4.
12 ÷ 4 = 3 and 16 ÷ 4 = 4, so 12/16 = 3/4. The GCF of 3 and 4 is 1, so 3/4 is in simplest form.
You can also divide by any common factor you spot, such as 2, and repeat until nothing more divides. You get the same answer, it just takes more steps.
Finding the GCF with Euclid's algorithm
Listing factors gets slow for big numbers. Euclid's algorithm, more than 2,000 years old, is much faster: divide the larger number by the smaller one, keep the remainder, and repeat with the smaller number and the remainder. When the remainder reaches 0, the last divisor is the GCF.
For 1071/462:
1071 = 2 × 462 + 147
462 = 3 × 147 + 21
147 = 7 × 21 + 0
The remainder is now 0, so the GCF is 21. Dividing both parts by 21 gives 1071 ÷ 21 = 51 and 462 ÷ 21 = 22, so 1071/462 = 51/22. The calculator shows these lines for every fraction you enter.
Simplifying improper fractions and mixed numbers
An improper fraction has a top number at least as big as the bottom, like 45/10. It simplifies in exactly the same way: the GCF is 5, so 45/10 = 9/2. To write it as a mixed number, divide: 9 ÷ 2 = 4 remainder 1, so 9/2 = 4 1/2.
For a mixed number such as 2 4/6, first turn it into an improper fraction: multiply the whole number by the denominator and add the numerator, giving (2 × 6 + 4)/6 = 16/6. The GCF of 16 and 6 is 2, so it simplifies to 8/3, which is 2 2/3. You could also simplify just the fraction part, 4/6 = 2/3, and keep the 2.
Equivalent fractions explained
Equivalent fractions look different but have the same value. You make one by multiplying, or dividing, the top and bottom by the same number: 3/4 = 6/8 = 9/12 = 12/16. Simplifying is just finding the equivalent fraction with the smallest numbers. Equivalent fractions are also how you give fractions a common denominator before adding or comparing them.
Common mistakes
- Dividing only one part. 12/16 is not 3/16 or 12/4. Whatever you divide the top by, you must divide the bottom by too.
- Cancelling digits. You can't cross out matching digits: crossing out the 6s in 16/64 happens to give the right answer, 1/4, but crossing out the 3s in 13/32 gives 1/2, and 13/32 is already in lowest terms. Always divide by a common factor.
- Stopping too early. Dividing 24/36 by 2 gives 12/18, which still simplifies. Dividing by the GCF (12) gets you straight to 2/3.
- Mixing up the sign. A negative fraction has one minus sign. If both the top and bottom are negative, the fraction is positive.
Frequently asked questions
How do you simplify a fraction?
Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 12/16 the GCF is 4, and 12 ÷ 4 = 3 and 16 ÷ 4 = 4, so 12/16 simplifies to 3/4.
How do I know a fraction is in simplest form?
A fraction is in simplest form (lowest terms) when the only whole number that divides both the numerator and the denominator is 1. In other words, their GCF is 1, as with 7/13 or 3/4.
Can you simplify an improper fraction?
Yes. Simplify it the same way, by dividing the top and bottom by their GCF: 45/10 becomes 9/2. You can then write it as a mixed number if you prefer, here 4 1/2.
Where does the minus sign go in a negative fraction?
−18/24, 18/−24 and −(18/24) are all the same number. By convention the simplified answer puts the minus sign in front or on the numerator, so all three simplify to −3/4.
Can I simplify a fraction that contains decimals?
Yes. First multiply the top and bottom by the same power of 10 to remove the decimals, then simplify as usual. For 2.5/5, multiplying both by 10 gives 25/50, which simplifies to 1/2.